arxiv.orgarXiv:0806.2963v1 [math.ST] 18 Jun 2008 The Annals of Statistics 2008, Vol. 36, No. 3, 1261–1298 DOI: 10.1214/07-AOS508 c Institute of Mathematical Statistics, 2008 OPTIMAL

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The Annals of Statistics

2008, Vol. 36, No. 3, 1261–1298DOI: 10.1214/07-AOS508c© Institute of Mathematical Statistics, 2008

OPTIMAL RANK-BASED TESTS FOR HOMOGENEITY

OF SCATTER

By Marc Hallin1 and Davy Paindaveine2

Universite Libre de Bruxelles

We propose a class of locally and asymptotically optimal tests,based on multivariate ranks and signs for the homogeneity of scattermatrices in m elliptical populations. Contrary to the existing para-metric procedures, these tests remain valid without any moment as-sumptions, and thus are perfectly robust against heavy-tailed dis-tributions (validity robustness). Nevertheless, they reach semipara-metric efficiency bounds at correctly specified elliptical densities andmaintain high powers under all (efficiency robustness). In particular,their normal-score version outperforms traditional Gaussian likeli-hood ratio tests and their pseudo-Gaussian robustifications under avery broad range of non-Gaussian densities including, for instance,all multivariate Student and power-exponential distributions.

1. Introduction.

1.1. Homogeneity of variances and covariance matrices. The assumptionof variance homogeneity is central to the theory and practice of univariatem-sample inference, playing a major role in such models as m-sample loca-tion (ANOVA) or m-sample regression (ANOCOVA). The problem of test-ing the null hypothesis of variance homogeneity, therefore, is of fundamentalimportance, and for more than half a century, has been a subject of contin-ued interest in the statistical literature. The standard procedure, describedin most textbooks, is Bartlett’s modified (Gaussian) likelihood ratio test(see [2]). This test, however, is well known to be highly nonrobust againstviolations of Gaussian assumptions, a fact that gave rise to a large number

Received May 2007; revised May 2007.1Supported by the Bendheim Center.2Supported by a Credit d’Impulsion of the Belgian Fonds National de la Recherche

Scientifique.AMS 2000 subject classifications. 62M15, 62G35.Key words and phrases. Elliptic densities, scatter matrix, shape matrix, local asymp-

totic normality, semiparametric efficiency, adaptivity.

This is an electronic reprint of the original article published by theInstitute of Mathematical Statistics in The Annals of Statistics,2008, Vol. 36, No. 3, 1261–1298. This reprint differs from the original inpagination and typographic detail.

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2 M. HALLIN AND D. PAINDAVEINE

of “robustified” versions of the likelihood ratio procedure ([3, 5, 6, 21], toquote only a few). Soon, it was noticed that these “robustifications,” if rea-sonably resistant to nonnormality, unfortunately were lacking power: in theconvenient terminology of Heritier and Ronchetti [22], they enjoy validityrobustness but not efficiency robustness.

In an extensive simulation study, Conover, Johnson and Johnson [7] haveinvestigated the validity robustness (against nonnormal densities) and effi-ciency robustness properties of 56 distinct tests, including several (signed-)rank-based ones. Their conclusion is that only three of them survive the ex-amination, and that two of the three survivors are normal-score signed-ranktests (adapted from [10]).

In view of its applications in MANOVA, MANOCOVA, discriminant anal-ysis, and so forth, the multivariate problem of testing for homogeneity ofcovariance matrices is certainly no less important than its univariate coun-terpart. The same problem moreover is of intrinsic interest in such fields aspsychometrics or genetics where, for instance, the homogeneity of geneticcovariance structure among species is a classical subject of investigation.Robustness and power issues, however, are even more delicate and complexin the multivariate context.

Here again, the traditional procedure is a Gaussian modified likelihood ra-

tio test, φ(n)MLRT, the unbiasedness of which has been established (for general

dimension k and number m of populations, under Gaussian assumptions)by Perlman [31]. Unfortunately, this test shares the poor resistance to non-normality of its univariate counterpart, and is invalid even under ellipticaldensities with finite fourth-order moments: see [12, 38] or [41]. The testproposed by Nagao [27], as shown in [19], is asymptotically equivalent to

φ(n)MLRT—still under finite fourth-order moments, and hence inherits of the

poor robustness properties of the latter. Quite surprisingly, and except for

some attempts to bootstrap φ(n)MLRT ([11, 42, 43]), the important problem

of testing homogeneity of covariance matrices under possibly non-Gaussiandensities has remained an open problem for more than fifty years.

In 2001, Schott [35] proposed aWald test, φ(n)Schott, the validity of which still

requires Gaussian densities, but also two modified versions thereof—denote

them as φ(n)Schott∗ and φ

(n)Schott†—enjoying validity robustness at homokurtic

and heterokurtic elliptical densities with finite fourth-order moments, re-spectively. More recently, a detailed account of the pseudo-Gaussian ap-proach of the problem has been given in [19], where we derive the locally

asymptotically most stringent (in the Le Cam sense) Gaussian test φ(n)N , and

show how to turn it into pseudo-Gaussian tests φ(n)N∗ and φ

(n)N† that are valid

under homokurtic and heterokurtic m-tuples of elliptical densities with fi-

nite fourth-order moments, respectively. We also show that φ(n)Schott∗ and φ

(n)N∗

RANK TESTS FOR SCATTER HOMOGENEITY 3

(resp. φ(n)Schott† and φ

(n)N†) asymptotically coincide. Schott himself recommends

using the homokurtic version φ(n)Schott∗ of his test and, since heterokurtic sam-

ples are not considered here, we will concentrate on φ(n)Schott∗ and φ

(n)N∗ as the

only pseudo-Gaussian tests available so far for the problem under study(one of the findings of [19] indeed is that the bootstrapped MLRT, althoughperfectly valid, is highly unsatisfactory in the non-Gaussian case).

This pseudo-Gaussian qualification, however, still requires finite momentsof order four, and only addresses validity robustness issues. Although nomultivariate equivalent of the Conover, Johnson and Johnson study [7] hasbeen conducted so far, the results we are obtaining in Sections 5.2 and 6

below, however, indicate that φ(n)Schott∗ and φ

(n)N∗ still suffer a severe lack of

efficiency robustness, particularly so at radial densities with high kurtosissuch as the Student with 4 + δ degrees of freedom (with δ ≈ 0).

It is thus very likely that the conclusions of Conover et al. [7] still ap-ply, which strongly suggests considering a “rank-based” approach—with aconcept of “ranks” adapted to the multivariate context. The purpose of thispaper is to develop such an approach under elliptical assumptions, based onthe signs and ranks considered in [15] for location, [16] for (auto)regression,[14] and [17] for shape.

Contrary to all existing methods, our tests do not require any momentassumptions, so that the null hypothesis they address actually is that ofhomogeneity of scatter matrices, reducing to more classical homogeneity ofcovariance matrices under finite second-order moments. They are asymptot-ically distribution-free, reach semiparametric efficiency at correctly specifieddensities, and are both validity- and efficiency-robust. When based on Gaus-sian scores, their asymptotic relative efficiency with respect to the Gaussianand pseudo-Gaussian tests is larger than one under almost all elliptical den-sities (see Section 6 for details).

1.2. Testing equality of scatter (covariance) matrices. Denote by (Xi1, . . . ,Xini), i= 1, . . . ,m, a collection of m mutually independent samples of i.i.d.k-dimensional random vectors with elliptically symmetric densities. Moreprecisely, for all i= 1, . . . ,m, the ni observations Xij , j = 1, . . . , ni, are as-sumed to have a probability density function of the form

fi(x) := ck,f1 |Σi|−1/2f1(((x− θi)′Σ−1

i (x− θi))1/2), x ∈R

k,(1.1)

for some k-dimensional vector θi (location), symmetric and positive definite(k× k) matrix Σi (the scatter matrix), and (duly standardized: see below)function f1 :R

+0 → R

+ (the radial density). The null hypothesis consideredthroughout is the hypothesis H0 :Σ1 = · · ·=Σm of scatter homogeneity (re-ducing, under finite variances, to covariance homogeneity).


Define (throughout, Σ1/2 stands for the symmetric root of Σ)

Uij(θi,Σi) :=Σ

−1/2i (Xij − θi)

‖Σ−1/2i (Xij − θi)‖

and

(1.2)dij(θi,Σi) := ‖Σ−1/2

i (Xij − θi)‖.Denoting by n=

∑mi=1 ni the total sample size and writingRij(θ1, . . . ,θm,Σ1,

. . . ,Σm) for the rank of dij(θi,Σi) among d11(θ1,Σ1), . . . , dmnm(θm,Σm),consider the signed-rank scatter matrices

S˜K;i :=

1

ni

ni∑

j=1

K

(Rij(θ1, . . . , θm, Σ, . . . , Σ)

n+1

)Uij(θi, Σ)U′

ij(θi, Σ),

(1.3)i= 1, . . . ,m,

where θ1, . . . , θm are consistent estimates of the various location parameters,Σ is a consistent (under H0) estimate of the common null value of the Σi’s,and K is some appropriate score function. The proposed signed-rank testsreject the null hypothesis of scatter homogeneity for large values of

Q˜(n)K :=

1

n

∑

1<i<i′<m

(ni + ni′)Q˜(n)K;i,i′,(1.4)

where Q˜

(n)K;i,i′ :=

nini′

ni+ni′{αk,K tr[(S

˜K;i−S

˜K;i′)

2]+βk,K tr2[S˜K;i−S

˜K;i′ ]}; αk,K

and βk,K are constants depending on the dimension k and the score func-tion K. The form of (1.4) follows from Le Cam type optimality arguments,

but Q˜

(n)K also can be obtained by replacing traditional sample covariance

matrices with the signed-rank scatter matrices (1.3) in the statistic of the

pseudo-Gaussian test φ(n)N∗ derived in [19].

The use of signed ranks is justified by the invariance principle: H0 in-deed is not only invariant under affine transformations, but also under thegroup of (continuous monotone) radial transformations; see Section 3.2 fordetails. Beyond affine-invariance (all tests considered in this paper are affine-invariant), our rank tests—unlike their competitors—are also (asymptoti-cally) invariant with respect to the groups of radial transformations; theirvalidity robustness actually follows from this latter invariance property.

As announced, our methodology combines validity and efficiency robust-ness. We will show that for (essentially) any radial density f1, it is possibleto define a score function K :=Kf1 characterizing a signed-rank test which islocally and asymptotically optimal (locally and asymptotically most stringent,in the Le Cam sense) under radial density f1. In particular, when based on


Gaussian scores, our rank tests achieve the same asymptotic performances

as the optimal Gaussian tests φ(n)N at the multinormal, while enjoying the

validity robustness of the pseudo-Gaussian φ(n)N∗ (or φ

(n)Schott∗) and even more,

since no moment assumption is required. Moreover, the asymptotic relativeefficiencies (AREs) of these normal-score tests are almost always larger thanone with respect to their parametric competitors; see Section 6. The class oftests we are proposing thus in most cases dominates the existing parametricones, both in terms of robustness and power.

1.3. Outline of the paper. The paper is organized as follows. In Section 2,we collect the main assumptions needed in the sequel. Section 3 discussessemiparametric modeling issues and their relation to group invariance. Sec-tion 4.1 states the uniform local asymptotic normality result (ULAN) onwhich our construction of locally and asymptotically optimal tests is based.In Section 4.2, we construct rank-based versions of the central sequencesappearing in this ULAN result. In Section 5.1, we derive and study the pro-posed nonparametric (signed-rank) tests [based on (1.4)] and, in Section 5.2,for the purpose of performance comparisons, the optimal pseudo-Gaussianones. Asymptotic relative efficiencies with respect to those pseudo-Gaussiantests are derived in Section 6. Section 7 provides some simulation results con-firming the theoretical ones. Finally, the Appendix collects proofs of asymp-totic linearity and other technical results.

2. Main assumptions. For the sake of convenience, we are collecting herethe main assumptions to be used in the sequel.

2.1. Elliptical symmetry. As mentioned before, we throughout assumethat all populations are elliptically symmetric. More precisely, defining thecollections F of radial densities and F1 of standardized radial densities as

F := {f > 0 a.e. :µk−1;f <∞}

and

F1 :=

{f1 ∈F :

1

µk−1;f1

∫ 1

0rk−1f1(r)dr =

1

2

},

respectively, where µℓ;f :=∫∞0 rℓf(r)dr, we require the following.

Assumption (A). The observations Xij , j = 1, . . . , ni, i= 1, . . . ,m, aremutually independent, with p.d.f. fi, i = 1, . . . ,m, given in (1.1), for somef1 ∈ F1.


Clearly, for the scatter matrices Σi in (1.1) to be well defined, identifiabil-ity restrictions are needed. This is why we impose that f1 ∈ F1, which impliesthat dij(θi,Σi) defined in (1.2) has median one under (1.1), and identifiesΣi without requiring any moment assumptions (see [17] for a discussion).Note, however, that under finite second-order moments, Σi is proportionalto the covariance Σ0i of Xij , with a proportionality constant that does notdepend on i: the hypotheses of scatter and covariance homogeneity thencoincide.

Special instances of elliptical densities are the k-variate multinormal dis-tributions, with standardized radial density f1(r) = φ1(r) := exp(−akr

2/2),the k-variate Student distributions, with radial densities (for ν ∈ R

+0 de-

grees of freedom) f1(r) = f t1,ν(r) := (1+ak,νr

2/ν)−(k+ν)/2, and the k-variatepower-exponential distributions, with radial densities of the form f1(r) =f e1,η(r) := exp(−bk,ηr

2η), η ∈R+0 ; the positive constants ak, ak,ν and bk,η are

such that f1 ∈ F1.The equidensity contours associated with (1.1) are hyper-ellipsoids cen-

tered at θi, the shape and orientation of which are determined by the scat-ter matrix Σi. The multivariate signs Uij(θi,Σi) and standardized radialdistances dij(θi,Σi) defined in (1.2) are the (within-group) elliptical coordi-nates associated with those ellipsoids: the observation Xij then decomposes

into θi + dijΣ1/2i Uij , where the Uij ’s, j = 1, . . . , ni, i = 1, . . . ,m are i.i.d.

uniform over the unit sphere in Rk, and the dij ’s are i.i.d., independent

of the Uij ’s, with common density f1k(r) := (µk−1;f1)−1rk−1f1(r) over R

+

(justifying the terminology standardized radial density for f1) and distribu-tion function F1k. In the sequel, the notation g1k and G1k will be used forthe corresponding functions computed from a standardized radial densityg1(∈ F1).

The derivation of locally and asymptotically optimal tests at radial den-sity f1 will be based on the uniform local and asymptotic normality (ULAN)of the model at given f1. This ULAN property—the statement of which re-quires some further preparation and is delayed to Section 4.1—only holdsunder some further mild regularity conditions on f1. More precisely, ULAN(see Proposition 4.1 below) requires f1 to belong to the collection Fa ofabsolutely continuous densities in F1 such that, letting ϕf1 :=−f1/f1 (with

f1 the a.e.-derivative of f1), the integrals

Ik(f1) :=∫ 1

0ϕ2f1(F

−11k (u))du and Jk(f1) :=

∫ 1

0ϕ2f1(F

−11k (u))(F−1

1k (u))2 du

are finite. The quantities Ik(f1) and Jk(f1) play the roles of radial Fisherinformation for location and radial Fisher information for shape/scale, re-spectively (see [17]).


2.2. Asymptotic behavior of sample sizes. Although, for the sake of no-tational simplicity, we do not mention it explicitly, we actually considersequences of statistical experiments, with triangular arrays of observations

of the form (X(n)11 , . . . ,X

(n)

1n(n)1

,X(n)21 , . . . ,X

(n)

2n(n)2

, . . . ,X(n)m1, . . . ,X

(n)

mn(n)m

) indexed

by the total sample size n, where the sequences n(n)i satisfy the following

assumption.

Assumption (B). For all i= 1, . . . ,m, ni = n(n)i →∞ as n→∞.

Note that this assumption is weaker than the corresponding classical as-sumption in (univariate or multivariate) multisample problems, which re-quires that ni/n be bounded away from 0 and 1 for all i as n→∞. Letting

λ(n)i := n

(n)i /n, it is easy to check that Assumption (B) is actually equivalent

to the Noether conditions

max

(1− λ

(n)i

λ(n)i

,λ(n)i

1− λ(n)i

)= o(n) as n→∞, for all i,

that are needed for the representation result in Lemma 4.1(i) below. How-ever, in the derivation of asymptotic distributions under local alternatives,the following reinforcement of Assumption (B) is assumed to hold (mainly,for notational comfort):

Assumption (B′). For all i= 1, . . . ,m, λ(n)i → λi ∈ (0,1), as n→∞.

2.3. Score functions. The score functions K appearing in the rank-basedstatistics (1.3)–(1.4) will be assumed to satisfy the following regularity con-ditions.

Assumption (C). The score function K : (0,1) → R (C1) is a contin-uous, nonconstant, and square-integrable mapping which (C2) can be ex-pressed as the difference of two monotone increasing functions, and (C3)

satisfies∫ 10 K(u)du= k.

Assumption (C3) is a normalization constraint that is automatically satis-fied by the score functions K(u) =Kf1(u) := ϕf1(F

−11k (u))F−1

1k (u) leading tolocal and asymptotic optimality at radial density f1 (at which ULAN holds);see Section 4.1. For score functions K,K1,K2 satisfying Assumption (C),let Jk(K1,K2) := E[K1(U)K2(U)] and Lk(K1,K2) := Cov[K1(U),K2(U)] =Jk(K1,K2)− k2 [throughout, U stands for a random variable uniformly dis-tributed over (0,1)]. For simplicity, we write Jk(K) for Jk(K,K), Lk(K) for


Lk(K,K), Jk(K,f1) for E[K(U)Kf1(U)], Lk(f1, g1) for E[Kf1(U)Kg1(U)]−k2, etc.

The power score functions Ka(u) := k(a+1)ua (a > 0) provide some tra-ditional score functions satisfying Assumption (C), with Jk(Ka) = k2(a+1)2/(2a + 1) and Lk(Ka) = k2a2/(2a + 1): Wilcoxon and Spearman scoresare obtained for a = 1 and a = 2, respectively. As for score functions ofthe form Kf1 , an important particular case is that of van der Waerdenor normal scores, obtained for f1 = φ1. Then, denoting by Ψk the chi-square distribution function with k degrees of freedom, Kφ1(u) = Ψ−1

k (u),Jk(φ1) = k(k + 2), and Lk(φ1) = 2k. Similarly, writing Gk,ν for the Fisher–Snedecor distribution function with k and ν degrees of freedom, Student den-sities f1 = f t

1,ν yield Kf t1,ν

(u) = k(k + ν)G−1k,ν(u)/(ν + kG−1

k,ν(u)), Jk(ft1,ν) =

k(k +2)(k + ν)/(k+ ν +2) and Lk(ft1,ν) = 2kν/(k + ν +2).

3. Semiparametric modeling of the family of elliptical densities.

3.1. Scatter, scale, and shape. Consider an observed n-tuple X1, . . . ,Xn

of i.i.d. k-dimensional elliptical random vectors, with location θ, scatter Σ,and radial density f1 ∈F1 but otherwise unspecified. The family P(n) of dis-tributions for this observation is indexed by (θ,Σ, f1). As soon as a semipara-metric point of view is adopted, or when rank-based methods are considered,the scatter matrix Σ naturally decomposes into Σ= σ2V, where σ is a scaleparameter (equivariant under multiplication by a positive constant) and V

a shape matrix (invariant under multiplication by a positive constant). Asemiparametric model with specified σ and unspecified standardized radialdensity f1 indeed would be highly artificial, and we, therefore, only considerthe case under which σ and f1 are both unspecified. This semiparametricsetting is also the one that enjoys the group invariance structure in whichthe ranks and the signs to be used in our method spontaneously arise frominvariance arguments; see Section 3.2 below.

The concepts of scale and shape however require a more careful definition.Denoting by Sk the collection of all k × k symmetric positive definite realmatrices, consider a function S :Sk → R

+0 satisfying S(λΣ) = λS(Σ) for all

λ ∈ R+0 and Σ ∈ Sk, and define scale and shape as σS := (S(Σ))1/2 and

VS := Σ/S(Σ), respectively. Clearly, VS is the only matrix in Sk whichis proportional to Σ and satisfies S(VS) = 1: denote by VS

k := {V ∈ Sk :S(V) = 1} the set of all possible shape matrices associated with S. Classicalchoices of S are (i) S(Σ) = (Σ)11 ([14, 17, 23] and [33]); (ii) S(Σ) = k−1 tr(Σ)([8, 28] and [39]); (iii) S(Σ) = |Σ|1/k ([9, 34, 36] and [37]).

In practice, all choices of S are essentially equivalent. Bickel (Example 4of [4], with a trace-based normalization of Σ−1) actually shows that irre-spective of S, the asymptotic information matrix for VS in the presence


of unspecified θ and σS is the same, at any θ ∈ Rk, σS ∈ R

+0 , VS ∈ VS

k

and f1, whether f1 is specified (parametric model) or not (semiparametricmodel): once θ and σS are unspecified, an unspecified f1 does not induceany additional loss for inference about VS . In [30], Paindaveine establishesthe stronger result that the information matrix for VS in the presence ofunspecified θ, σS and f1 is strictly less, at any θ ∈ R

k, σS ∈ R+0 , VS ∈ VS

k

and f1, than in the corresponding parametric model with specified θ, σSand f1—except for S :Σ 7→ |Σ|1/k, where those two information matricescoincide: under this determinant-based normalization, thus, the presence ofnuisances (θ, σS and f1) (resp., θ, VS , and f1) asymptotically has no effecton inference about shape (resp., inference about scale). In both cases, it canbe said (adopting a point estimation terminology) that shape can be esti-mated adaptively. This Paindaveine adaptivity, however, where θ, σS and f1lie in the nuisance space of the semiparametric model, is much stronger thanBickel adaptivity where only f1 does. This finding strongly pleads in favorof the determinant-based definition of shape which, with its block-diagonalinformation matrix for θ, σS and VS , is also more convenient from the pointof view of statistical inference, and as we will see in Section 5.1, allows foran ANOVA-type decomposition of the test statistics into two mutually in-dependent parts providing tests against subalternatives of scale and shapeheterogeneity, respectively. Therefore, throughout we adopt S(Σ) = |Σ|1/k,and henceforth simply write V ∈ Vk and σ for the resulting shape and scale.

The parameter in our problem then is the L-dimensional vector

ϑ := (ϑ′I ,ϑ

′II,ϑ′

III)′ := (θ′

1, . . . ,θ′m, σ2

1 , . . . , σ2m, (

◦vechV1)

′, . . . , (◦

vechVm)′)′,

where L=mk(k+3)/2 and◦

vech(V) is characterized by vech(V) =: ((V)11,

(◦

vechV)′)′. Indeed, Σi is entirely determined by σ2i and

◦vech(Vi). Write Θ

for the set Rmk × (R+0 )

m × (◦

vech(Vk))m of admissible ϑ values, and P

(n)ϑ;f1

or

P(n)ϑI ,ϑII ,ϑIII ;f1

for the joint distribution of the n observations under parameter

value ϑ and standardized radial density f1 (always implicitly assumed tobelong to F1, when notation f1 is used).

Finally, note that (i) Uij(θi,Σi) = Uij(θi,Vi) and (ii) dij(θi,Σi) =σ−1i dij(θi,Vi). It follows from (ii) that under H0, the ranks of the radial

distances computed from the common value V of the shape matrices co-incide with those of the standardized radial distances computed from thecommon value Σ of the scatter matrices.

3.2. Invariance issues. Denoting by M(Υ) the vector space spanned bythe columns of the L× r full-rank matrix Υ (r < L), the null hypothesis


of scatter homogeneity H0 : σ21V1 = · · ·= σ2

mVm can be written as H0 :ϑ ∈M(Υ), with

Υ :=

ΥI 0 0

0 ΥII 0

0 0 ΥIII

:=

Imk 0 0

0 1m 0

0 0 1m ⊗ Ik0

,

(3.1)

k0 :=k(k+ 1)

2− 1,

where 1m := (1, . . . ,1)′ ∈Rm and Iℓ denotes the ℓ-dimensional identity ma-

trix.Two distinct invariance structures play a role here. The first one is related

with the group of affine transformations of the observations, which generates

the parametric families P(n)Υ,f1

:=⋃

ϑ∈M(Υ){P(n)ϑ;f1

}. More precisely, this group

is the group Gm,k,◦ of affine transformations of the form Xij 7→AXij + bi,where A is a full-rank (k × k) matrix and B := (b1, . . . ,bm) a (k × m)matrix. Associated with that group is the group Gm,k,◦ of transformations

ϑ 7→ gm,kA,B(ϑ) of the parameter space, where

gm,kA,B(ϑ) := ((Aθ1 + b1)

′, . . . , (Aθm +bm)′, |A|2/kσ21 , . . . , |A|2/kσ2

m,

(◦

vech(AV1A′))′/|A|2/k, . . . , ( ◦

vech(AVmA′))′/|A|2/k)′.

Clearly, H0 is invariant under Gm,k,◦—meaning that gm,kA,B(M(Υ)) =M(Υ)

for all gm,kA,B. Therefore, it is reasonable to restrict to affine-invariant tests of

H0. Beyond their distribution-freeness with respect to the θi’s and the com-mon null values σ and V of the scale and shape parameters, affine-invarianttest statistics—that is, statistics Q such that Q(AX11 + b1, . . . ,AXmnm +bm) =Q(X11, . . . ,Xmnm) for allA,b1, . . . ,bm—yield tests that are coordinate-free.

A second invariance structure is induced by the groups G,◦ := GϑI ,V,◦ ofcontinuous monotone radial transformations, of the form

X 7→ Gg(X11, . . . ,Xmnm)

= Gg(θ1 + d11(θ1,V)V1/2U11(θ1,V), . . . ,

θm + dmnm(θm,V)V1/2Umnm(θm,V))

:= (θ1 + g(d11(θ1,V))V1/2U11(θ1,V), . . . ,

θm + g(dmnm(θm,V))V1/2Umnm(θm,V)),

where g :R+ →R+ is continuous, monotone increasing, and such that g(0) =

0 and limr→∞ g(r) =∞. For each ϑI ,V, this group GϑI ,V,◦ is a generating


group for P(n)ϑI ,V

:=⋃

σ

⋃f1{P

(n)

ϑI ,σ21m,1m⊗(◦

vechV);f1} (a nonparametric fam-

ily). In such families, the invariance principle suggests basing inference onstatistics that are measurable with respect to the corresponding maximal in-variant, namely the vectors (U11, . . . ,Umnm) of signs and the vector (R11, . . . ,Rmnm) of ranks, whereUij =Uij(θi,V), andRij =Rij(θ1, . . . ,θm,V, . . . ,V).

Such invariant statistics of course are distribution-free under P(n)ϑI ,V

.

4. Uniform local asymptotic normality, signs and ranks.

4.1. Uniform local asymptotic normality (ULAN ). As mentioned in Sec-tion 1, we plan to develop tests that are optimal at correctly specified den-sities, in the sense of Le Cam’s asymptotic theory of statistical experiments.In this section, we state the ULAN result (with respect to location, scaleand shape parameters for fixed radial density f1) on which optimality willbe based. Writing

ϑ(n) = (ϑ(n)′I ,ϑ

(n)′II

,ϑ(n)′III

)′

= (θ(n)′1 , . . . ,θ(n)′

m , σ2(n)1 , . . . , σ2(n)

m , (◦

vechV(n)1 )′, . . . , (

◦vechV(n)

m )′)′

for an arbitrary sequence of L-dimensional parameter values in Θ, considersequences of “local alternatives” ϑ(n) + n−1/2ν(n)τ (n), where

τ (n) = (τ(n)′I ,τ

(n)′II

,τ(n)′III

)′

= (t(n)′1 , . . . , t(n)′m , s

2(n)1 , . . . , s2(n)m , (

◦vechv

(n)1 )′, . . . , (

◦vechv(n)

m )′)′

is such that supn τ(n)′τ (n) < ∞ and where, denoting by Λ(n) = (Λ

(n)ii′ ) the

(m×m) diagonal matrix with Λ(n)ii := (λ

(n)i )−1/2 (see Section 2.2),

ν(n) :=

ν(n)I 0 0

0 ν(n)II

0

0 0 ν(n)III

:=

Λ(n) ⊗ Ik 0 0

0 Λ(n) 0

0 0 Λ(n) ⊗ Ik0

(4.1)

[under Assumption (B′), we also write ν for limn→∞ν(n)]. Clearly, these

local alternatives do not involve (v(n)i )11, i= 1, . . . ,m. It is natural, though,

to see that the perturbed shapes V(n)i +n

−1/2i v

(n)i remain [up to o(n

−1/2i )’s]

within the family Vk of shape matrices: this leads to defining (v(n)i )11 in

such a way that tr[(V(n)i )−1v

(n)i ] = 0, i = 1, . . . ,m, which entails |V(n)

i +

n−1/2i v

(n)i |1/k = 1+ o(n

−1/2i ) (see [18], Section 4).

The following notation will be used throughout. Let diag(B1,B2, . . . ,Bm)stand for the block-diagonal matrix with diagonal blocks B1, B2, . . . ,Bm.


Write V⊗2 for the Kronecker product V⊗V. Denoting by eℓ the ℓth vectorof the canonical basis of Rk, let Kk :=

∑ki,j=1(eie

′j)⊗ (eje

′i) be the (k

2× k2)commutation matrix, and put Jk := (vec Ik)(vec Ik)

′. Finally, let Mk(V) be

the (k0 × k2) matrix such that (Mk(V))′(◦

vechv) = (vecv) for any symmet-ric k×k matrix v such that tr(V−1v) = 0. As shown in Lemma 4.2(v) of [30],Mk(V)(vecV−1) = 0 for all V ∈ Vk.

We then have the following ULAN result; the proof follows along the samelines as in Theorem 2.1 of [30], and hence is omitted.

Proposition 4.1. Assume that (A) and (B) hold, and that f1 ∈ Fa.

Then the family P(n)f1

:= {P(n)ϑ;f1

|ϑ ∈Θ} is ULAN, with central sequence

∆ϑ;f1 =∆(n)ϑ;f1

:=

∆I

ϑ;f1

∆II

ϑ;f1

∆III

ϑ;f1

,

∆Iϑ;f1 =

∆I,1ϑ;f1...

∆I,mϑ;f1

,

∆II

ϑ;f1 =

∆II ,1ϑ;f1...

∆II ,mϑ;f1

, ∆III

ϑ;f1 =

∆III ,1ϑ;f1...

∆III ,mϑ;f1

,

where [with dij = dij(θi,Vi) and Uij =Uij(θi,Vi)]

∆I,iϑ;f1

:=n−1/2i

σi

ni∑

j=1

ϕf1

(dijσi

)V

−1/2i Uij,

∆II ,iϑ;f1

:=n−1/2i

2σ2i

ni∑

j=1

(ϕf1

(dijσi

)dijσi

− k

),

∆III ,iϑ;f1

:=n−1/2i

2Mk(Vi)(V

⊗2i )−1/2

ni∑

j=1

ϕf1

(dijσi

)dijσi

vec(UijU′ij),

i= 1, . . . ,m, and full-rank block-diagonal information matrix

Γϑ;f1 := diag(ΓIϑ;f1 ,Γ

II

ϑ;f1 ,ΓIII

ϑ;f1),(4.2)

where, defining σ := diag(σ1, . . . , σm), V := diag(V1, . . . ,Vm), Mk(V) :=diag(Mk(V1), . . . ,Mk(Vm)) and V⊗2 := diag(V⊗2

1 , . . . ,V⊗2m ), we let

ΓIϑ;f1 :=

1

kIk(f1)(σ−2 ⊗ Ik)V

−1, ΓII

ϑ;f1 :=14Lk(f1)σ

−4


and

ΓIII

ϑ;f1 :=Jk(f1)

4k(k+ 2)Mk(V)[Im ⊗ (Ik2 +Kk)](V

⊗2)−1(Mk(V))′.

More precisely, for any ϑ(n) = ϑ+O(n−1/2) and any bounded sequence τ (n),we have

Λ(n)

ϑ(n)+n−1/2ν(n)τ(n)/ϑ(n);f1

:= log(dP(n)

ϑ(n)+n−1/2ν(n)τ(n);f1

/dP(n)

ϑ(n);f1

)

= (τ (n))′∆(n)

ϑ(n);f1

− 12(τ

(n))′Γϑ;f1τ(n) + oP(1)

and ∆ϑ(n);f1

L−→N (0,Γϑ;f1) under P(n)

ϑ(n);f1

, as n→∞.

The classical theory of hypothesis testing in Gaussian shifts (see Section11.9 of [26]) provides the general form for locally asymptotically optimal(namely, most stringent) tests in ULAN models. Such tests, for a null hy-pothesis of the form ϑ ∈ M(Υ), should be based on the asymptoticallychi-square null distribution of

QΥ :=∆′ϑ;f1Γ

−1/2ϑ;f1

[I− proj(Γ1/2ϑ;f1

(ν(n))−1Υ)]Γ

−1/2ϑ;f1

∆ϑ;f1

[with ϑ replaced by an appropriate estimator ϑ; see Assumption (D) below],where proj(A) =A[A′A]−1A′, for any (L× r) matrix A with full rank r, isthe matrix projecting R

L onto M(A). Whenever Γϑ;f1 , ν(n) and Υ all hap-

pen to be block-diagonal, which is the case in our problem, this projectionmatrix clearly is block-diagonal, with diagonal

blocks proj((ΓIϑ;f1)

1/2(ν(n)I )−1ΥI), proj((ΓII

ϑ;f1)1/2(ν

(n)II

)−1ΥII ), and

proj((ΓIII

ϑ;f1)1/2(ν

(n)III

)−1ΥIII ), denoting projections in Rmk, Rm and R

mk0 ,

respectively. Since moreover M((ΓIϑ;f1)

1/2(ν(n)I )−1 × ΥI) = R

mk, we have

proj((ΓIϑ;f1)

1/2(ν(n)I )−1ΥI) = Imk, so that QΥ reduces to

QΥ = (∆II

ϑ;f1)′(ΓII

ϑ;f1)−1/2

× [I− proj((ΓII

ϑ;f1)1/2(ν

(n)II

)−1ΥII )](Γ

II

ϑ;f1)−1/2∆II

ϑ;f1(4.3)

+ (∆III

ϑ;f1)′(ΓIII

ϑ;f1)−1/2

× [I− proj((ΓIII

ϑ;f1)1/2(ν

(n)III

)−1ΥIII )](Γ

III

ϑ;f1)−1/2∆III

ϑ;f1 ,

where ∆Iϑ;f1 does not play any role. Accordingly, in the next section, we

proceed with rank-based analogues of ∆II

ϑ;f1 and ∆III

ϑ;f1 only.Note that the decomposition (4.3) of QΥ into two asymptotically orthog-

onal quadratic forms corresponds to the decomposition of scatter hetero-geneity into scale and shape heterogeneity; each of the two quadratic forms


in the right-hand side of (4.3) actually constitutes a locally asymptoticallyoptimal test statistic for H0 against one of those two subalternatives.

4.2. A rank-based central sequence for scale and shape (scatter). A gen-eral result by [20] implies that, in adaptive models for which fixed-f1 sub-models are ULAN and fixed-ϑ submodels are generated by a group Gϑ, in-variant versions of central sequences exist under very general assumptions.In the present context, this result would imply the existence, for the nullvalues of ϑ, of central sequences based on the multivariate signs Uij and theranks Rij . Although that result does not directly apply here, it is very likelythat it still holds. This fact is confirmed by the asymptotic representationof Lemma 4.1(i) below.

Consider the signed-rank statistic [associated with some score function K

satisfying Assumption (C)]∆˜ϑ;K := ((∆

˜II

ϑ;K)′, (∆˜III

ϑ;K)′)′ := (∆˜II ,1ϑ;K , . . . ,∆

˜II ,mϑ;K ,

(∆˜

III ,1ϑ;K )′, . . . , (∆

˜III ,mϑ;K )′)′, where

∆˜II ,iϑ;K :=

n−1/2i

2σ2i

ni∑

j=1

(K

(Rij

n+ 1

)− k

)(4.4)

and

∆˜

III ,iϑ;K :=

n−1/2i

2Mk(Vi)(V

⊗2i )−1/2

ni∑

j=1

K

(Rij

n+1

)vec(UijU

′ij).(4.5)

The following lemma provides (i) an asymptotic representation and (ii) theasymptotic distribution of ∆

˜ϑ;K (see the Appendix for the proof). An im-

mediate corollary of (i) is that ∆˜ϑ;f1 := ∆

˜ϑ;Kf1

, with K = Kf1 , actually

constitutes a signed-rank version of the scatter part ((∆II

ϑ;f1)′, (∆III

ϑ;f1)′)′ of

the central sequence ∆ϑ;f1 .

Lemma 4.1. Assume that (A), (B) and (C) hold. Fix ϑ ∈M(Υ) (withcommon values σ and V of the scale and shape parameters). Let Rij bethe rank of dij := dij(θi,V) among d11, . . . , dmnm , and let Uij :=Uij(θi,V).Then, for all g1 ∈ F1:

(i) ∆˜ϑ;K =∆ϑ;K;g1 + oL2(1), under P

(n)ϑ;g1

, as n→∞, where ∆ϑ;K;g1 :=

((∆II

ϑ;K;g1)′, (∆III

ϑ;K;g1)′)′ := (∆II ,1

ϑ;K;g1, . . . ,∆II ,m

ϑ;K;g1, (∆III ,1

ϑ;K;g1)′, . . . , (∆III ,m

ϑ;K;g1)′)′,

with

∆II ,iϑ;K;g1

:=n−1/2i

2σ2

ni∑

j=1

(K

(G1k

(dijσ

))− k

)


and

∆III ,iϑ;K;g1

:=n−1/2i

2Mk(V)(V⊗2)−1/2

ni∑

j=1

K

(G1k

(dijσ

))vec(UijU

′ij);(4.6)

(ii) defining Hk(V) := 14k(k+2)Mk(V)[Ik2 + Kk](V

⊗2)−1(Mk(V))′,

∆ϑ;K;g1 is asymptotically normal with mean zero and mean( 1

4σ4Lk(K,g1)τ II

Jk(K,g1)[Im ⊗Hk(V)]τ III

)

under P(n)ϑ;g1

and P(n)

ϑ+n−1/2ν(n)τ ;g1, respectively, and covariance matrix

Γϑ;K := diag(ΓII

ϑ;K ,ΓIII

ϑ;K) := diag

(1

4σ4Lk(K)Im,Jk(K)[Im ⊗Hk(V)]

)

under both (the claim under P(n)

ϑ+n−1/2ν(n)τ;g1further requires g1 ∈Fa).

As mentioned in the description of the most stringent tests (see the com-ments after Proposition 4.1), we will need replacing the parameter ϑ with

some estimate. For this purpose, we assume the existence of ϑ := ϑ(n) sat-isfying:

Assumption (D). The sequence of estimators (ϑ(n), n ∈N) is:

(D1) constrained : P(n)ϑ;g1

[ϑ(n) ∈M(Υ)] = 1 for all n, ϑ ∈M(Υ), and g1 ∈F1;

(D2) n1/2(ν(n))−1-consistent : for all ϑ ∈ M(Υ), n1/2(ν(n))−1(ϑ(n) − ϑ) =

OP(1), as n→∞, under⋃

g1∈F1{P(n)

ϑ;g1};

(D3) locally asymptotically discrete: for all ϑ ∈M(Υ) and all c > 0, there

exists M =M(c)> 0 such that the number of possible values of ϑ(n)

in balls of the form {t ∈RL :n1/2‖(ν(n))−1(t−ϑ)‖ ≤ c} is bounded by

M as n→∞, and(D4) affine-equivariant : denoting by ϑ(n)(A,B) the value of ϑ(n) computed

from the transformed sample AXij + bi, j = 1, . . . , ni, i = 1, . . . ,m,

ϑ(n)(A,B) = gm,kA,B(ϑ

(n)), for all gm,kA,B ∈ Gm,k.

There are many possible choices for ϑ. However, still in order to avoidmoment assumptions, we propose the following estimators, related with theaffine-equivariant median proposed by [23]. For each i= 1, . . . ,m, let θi and

Vi be characterized by

1

ni

ni∑

j=1

Uij(θi, Vi) = 0 and1

ni

ni∑

j=1

Uij(θi, Vi)U′ij(θi, Vi) =

1

kIk,


with |Vi| = 1. Then, under ϑ ∈ M(Υ), the common value V of the Vi’s

is consistently estimated [as n→∞, under⋃

g1∈F1{P(n)

ϑ;g1} and Assumptions

(A) and (B), without any moment assumption on g1], at the rate required by

Assumption (D2), by the Tyler [39] estimator V computed from the n data

points Xij − θi and normalized in such a way that |V|= 1. Under the sameconditions, the null value σ of the scale is the common median of the i.i.d.radial distances dij(θi,V), so that the empirical median σ of the dij(θi, V)’scan be used as an estimator of σ. Consequently, the estimator

ϑ := (θ′

1, . . . , θ′

m, σ21′m,1′m ⊗ (◦

vech V)′)′(4.7)

satisfies (D2) above—except perhaps for the ϑII part which, however, is notinvolved in the test statistics below. This estimator also satisfies (D1) and(D4). As for (D3), it is a purely technical requirement, with little practicalimplications (for fixed sample size, any estimator indeed can be consideredpart of a locally asymptotically discrete sequence). Therefore, we henceforthassume that (4.7) satisfies Assumption (D).

The resulting ranks Rij := Rij(θ1, . . . , θm, V, . . . , V) are usually calledaligned ranks. The following asymptotic linearity result describes the asymp-

totic behavior of the aligned version ∆˜ ϑ;K of ∆

˜ϑ;K under P

(n)ϑ;g1

; see the

Appendix for the proof.

Proposition 4.2. Assume that Assumptions (A), (B), (C) and (D1)–(D3) hold. Let g1 ∈Fa and ϑ ∈M(Υ) (with common values σ and V forthe scale and shape parameters). Then,

∆˜

II

ϑ;K−∆˜II

ϑ;K +1

4σ4Lk(K,g1)(ν

(n)II

)−1n1/2(ϑ(n)II

−ϑII )

and

∆˜III

ϑ;K−∆˜

III

ϑ;K +Jk(K,g1)[Im ⊗Hk(V)](ν(n)III

)−1n1/2(ϑ(n)III

−ϑIII )

are oP(1) under P(n)ϑ;g1

, as n→∞.

5. Optimal tests of scatter homogeneity.

5.1. Optimal rank-based tests. For all ϑ ∈M(Υ) (with common values σand V for the scale and shape parameters), define

PII

ϑ;K := (ΓII

ϑ;K)−1 − (ν(n)II

)−1ΥII (Υ

′II (ν

(n)II

)−1ΓII

ϑ;K(ν(n)II

)−1ΥII )

−1Υ′

II (ν(n)II

)−1

=4σ4

Lk(K)[Im −C(n)]


and

PIII

ϑ;K := (ΓIII

ϑ;K)−1 − (ν(n)III

)−1ΥIII (Υ

′III (ν

(n)III

)−1ΓIII

ϑ;K(ν(n)III

)−1ΥIII )

−1Υ′

III (ν(n)III

)−1

= (Jk(K))−1[Im −C(n)]⊗ (Hk(V))−1,

whereC(n) = (C(n)ii′ ) denotes the (m×m) matrix with entries C

(n)ii′ := (λ

(n)i λ

(n)i′ )1/2.

The K-score version φ˜(n)K of the rank-based tests we are proposing rejects

H0 :ϑ ∈M(Υ) whenever the (affine-invariant) test statistic

Q˜

(n)K := (∆

˜II

ϑ;K)′PII

ϑ;K∆˜

II

ϑ;K+ (∆

˜III

ϑ;K)′PIII

ϑ;K∆˜

III

ϑ;K

=m∑

i,i′=1

[δi,i′

ni− 1

n

]

×ni∑

j=1

ni′∑

j′=1

{1

Lk(K)

(K

(Rij

n+ 1

)− k

)(K

(Ri′j′

n+1

)− k

)(5.1)

+k(k +2)

2Jk(K)K

(Rij

n+1

)K

(Ri′j′

n+1

)

×((U′

ijUi′j′)2 − 1

k

)}

exceeds the α-upper quantile χ2(m−1)(k0+1);1−α of the chi-square distribution

with (m− 1)(k0+1) degrees of freedom (δi,i′ stands for the usual Kroneckersymbol); the explicit form of (Hk(V))−1 allowing for (5.1) can be found in

Lemma 5.2 of [18]. In the sequel, we write φ˜(n)f1

and Q˜

(n)f1

for φ˜(n)Kf1

and Q˜

(n)Kf1

,

respectively.

The decomposition (5.1) of the rank-based test statistic Q˜

(n)K into two

asymptotically orthogonal terms parallels the corresponding decomposition(4.3) of QΥ (see the closing remark of Section 4.1), with the same interpre-tation in terms of subalternatives of scale and shape heterogeneity, respec-tively.

We are now ready to state the main result of this paper; for the sakeof simplicity, asymptotic powers are expressed under Assumption (B′) andperturbations τ (n) such that limn→∞ν(n)τ (n) = ντ /∈M(Υ), with νII τ II =

(s21/√λ1, . . . , s

2m/

√λm)′ and νIII τ III = ((

◦vechv1)

′/√λ1, . . . , (

◦vechvm)′/

√λm)′.

For any such τ and any ϑ ∈M(Υ) (still with common values σ and V ofthe scale and shape parameters), let

rIIϑ,τ :=1

σ4limn→∞

{(τ (n)II

)′[Im −C(n)]τ(n)II

}


(5.2)

=∑

1≤i<i′≤m

λiλi′

σ4

(s2i√λi

− s2i′√λi′

)2

and

rIIIϑ,τ := 2k(k + 2) limn→∞

{(τ (n)III

)′[[Im −C(n)]⊗Hk(V)]τ(n)III

}(5.3)

=∑

1≤i<i′≤m

λiλi′ tr

[(V−1

(vi√λi

− vi′√λi′

))2];

note that tr(V−1vi) = 0 for all i (see the comments before Proposition 4.1).

Theorem 5.1. Assume that (A), (B), (C) and (D1)–(D3) hold. Then:

(i) Q˜(n)K is asymptotically chi-square with (m−1)(k0+1) degrees of free-

dom under⋃

ϑ∈M(Υ)

⋃g1∈Fa

{P(n)ϑ;g1

}, and [provided that (B) is reinforced into

(B′)] asymptotically noncentral chi-square, still with (m− 1)(k0 +1) degreesof freedom, but with noncentrality parameter

L2k(K,g1)

4Lk(K)rIIϑ,τ +

J 2k (K,g1)

2k(k +2)Jk(K)rIIIϑ,τ(5.4)

under P(n)

ϑ+n−1/2ν(n)τ(n);g1, ϑ ∈M(Υ), ντ := limn→∞ ν(n)τ (n) /∈M(Υ), and

g1 ∈Fa;

(ii) the sequence of tests φ˜(n)K has asymptotic level α under

⋃ϑ∈M(Υ)

⋃g1∈Fa

{P(n)ϑ;g1

};(iii) if f1 ∈ Fa and Kf1 satisfies Assumption (C), the sequence of tests

φ˜(n)f1

is locally and asymptotically most stringent, still at asymptotic level α,

for⋃

ϑ∈M(Υ)

⋃g1∈Fa

{P(n)ϑ;g1

} against alternatives of the form⋃

ϑ/∈M(Υ){P(n)ϑ;f1

}.

See the Appendix for the proof. After some algebra, one obtains

Q˜

(n)K =

1

n

∑

1≤i<i′≤m

(ni + ni′)Q˜

(n)K;i,i′,

where

Q˜

(n)K;i,i′ =

nini′

ni + ni′


×{

1

Lk(K)

[1

ni

ni∑

j=1

K

(Rij

n+1

)− 1

ni′

ni′∑

j′=1

K

(Ri′j′

n+ 1

)]2

+k(k +2)

2Jk(K)(5.5)

×∥∥∥∥∥

[1

ni

ni∑

j=1

K

(Rij

n+ 1

)vec

(UijU

′ij −

1

kIk

)]

−[1

ni′

ni′∑

j′=1

K

(Ri′j′

n+ 1

)vec

(Ui′j′U

′i′j′ −

1

kIk

)]∥∥∥∥∥

2}

is the test statistic obtained in the two-sample case (for populations i and i′);see [40] for a similar decomposition in MANOVA problems. As announced,

no estimate ϑII of the common scale appears in the test statistics. Also,letting

S˜K;i :=

1

ni

ni∑

j=1

K

(Rij

n+ 1

)UijU

′ij ,

the statistics in (5.5) take the simple form

Q˜

(n)K;i,i′ =

nini′

ni + ni′

{1

Lk(K)tr2[S

˜K;i −S

˜K;i′ ]

+k(k+ 2)

2Jk(K)

[tr[(S

˜K;i −S

˜K;i′)

2]− 1

ktr2[S

˜K;i −S

˜K;i′]

]}

=nini′

ni + ni′

{k(k +2)

2Jk(K)tr[(S

˜K;i −S

˜K;i′)

2]

− k(Jk(K)− k(k +2))

2Jk(K)Lk(K)tr2[S

˜K;i −S

˜K;i′]

}.

For Gaussian scores (i.e., for K =Kφ1 ; see Section 2.3), one obtains thevan der Waerden test statistics

Q˜

(n)vdW =

1

n

∑

1≤i<i′≤m

(ni + ni′)Q˜

(n)vdW;i,i′ ,(5.6)

where

Q˜

(n)vdW;i,i′ =

nini′

2(ni + ni′)tr[(S

˜vdW;i −S

˜vdW;i′)

2],


with S˜vdW;i := n−1

i

∑nij=1Ψ

−1k (Rij/(n+ 1))UijU

′ij . The Student scores (i.e.,

K =Kf t1,ν

; see Section 2.3 again) yield

Q˜

(n)f t1,ν

=1

n

∑

1≤i<i′≤m

(ni + ni′)Q˜

(n)f t1,ν ;i,i

′ ,(5.7)

where

Q˜

(n)f t1,ν ;i,i

′ =nini′

ni + ni′

k+ ν + 2

2(k + ν)

×{tr[(S

˜f t1,ν ;i

−S˜f t1,ν ;i

′)2] +1

νtr2[S

˜f t1,ν ;i

−S˜f t1,ν ;i

′ ]

}

with S˜f t1,ν ;i

:= k(k+ν)n−1i

∑nij=1G

−1k,ν(Rij/(n+1))/[ν+kG−1

k,ν(Rij/(n+1))]×

UijU′ij . As for the tests associated with the power score functions Ka, they

are based on

Q˜

(n)Ka

=1

n

∑

1≤i<i′≤m

(ni + ni′)Q˜

(n)Ka;i,i′

, a > 0,(5.8)

where, letting S˜Ka;i := k(a+ 1)(n+ 1)−an−1

i

∑nij=1(Rij)

aUijU′ij ,

Q˜

(n)Ka;i,i′

=nini′

ni + ni′

2a+1

2a2(a+1)2k2{a2k(k +2) tr[(S

˜Ka;i −S

˜Ka;i′)

2]

− (a2k− 4a− 2) tr2[S˜Ka;i −S

˜Ka;i′ ]}.

Corollary 5.1. Assume that the conditions of Theorem 5.1 hold. Then:

(i) provided that g1 ∈ Fa is such that Lk(K,g1) 6= 0 6= Jk(K,g1), φ˜(n)K is

consistent under any local g1-alternative [i.e., under any P(n)

ϑ+n−1/2ν(n)τ(n);g1,

ϑ ∈M(Υ), limn→∞ ν(n)τ (n) /∈M(Υ)];(ii) the same conclusion holds if u 7→ K(u) absolutely continuous with

a.e. derivative K, and if g1 ∈Fa is such that∫∞0 K(G1k(r))r(g1k(r))

2 dr > 0

(in particular, if K is nondecreasing).

We refer to the Appendix for the proof. This corollary shows that the vander Waerden tests above, as well as those achieving local asymptotic strin-gency at prespecified Student or power-exponential densities, are locally con-sistent against arbitrary elliptical alternatives, since the corresponding score


functions are strictly increasing. Similar conclusions hold for the tests asso-ciated with the power score functions Ka, a > 0. More general consistencyresults, against nonlocal and possibly nonelliptical alternatives, of course,are highly desirable, and can be obtained by exploiting Hajek’s results onrank statistics under alternatives (see [13]), much along the same lines asin Section 5.2 of [17]. Such results, which we do not include here, implythat consistency is achieved at “almost all” nonlocal alternatives—the ex-ception being those very particular densities achieving a set of orthogonalityconditions involving the scores and the Uij ’s.

Another important concern is validity under homogeneous scatter butpossibly heterogeneous or/and nonelliptical densities. The ranks of the dij ’sthen lose their distribution-freeness and/or their independence with respectto the Uij ’s. Under totally arbitrary situations, little can be said, but it isextremely unlikely that the tests we are developing here remain valid; if ellip-ticity and finite moments of order four can be assumed, the pseudo-Gaussianmethods developed in [19], which remain valid under possibly heterokurticelliptic densities, are preferable.

5.2. The optimal pseudo-Gaussian tests. As explained in the Introduction,

the pseudo-Gaussian tests φ(n)N∗ and φ

(n)Schott∗ are the natural competitors of

our rank-based procedures. Since they are asymptotically equivalent (see

[19]), we concentrate on φ(n)N∗. This test is valid under finite fourth-order mo-

ments Ek(g1) :=∫ 10 (G

−11k (u))

4 du = (µk−1;g1)−1∫∞0 rk+3g1(r)dr, that is, un-

der any P(n)ϑ;g1

such that g1 ∈ F41 := {g1 ∈ F1 :Ek(g1)<∞}. For all such g1,

let Dk(g1) :=∫ 10 (G

−11k (u))

2 du: then κk(g1) :=k

(k+2)Ek(g1)D2

k(g1)

− 1 is a measure of

kurtosis (see, e.g., page 54 of [1]), common to the m elliptic populations un-

der P(n)ϑ;g1

. This kurtosis is consistently (still under P(n)ϑ;g1

, g1 ∈ F41 ) estimated

by

κk :=k

(k+ 2)

(n−1∑mi=1

∑nij=1 d

4ij)

(n−1∑m

i=1

∑nij=1 d

2ij)

2− 1,

where dij := dij(Xi,Si), with Xi := n−1i

∑nij=1Xij and Si := n−1

i

∑nij=1(Xij −

Xi)(Xij − Xi)′. At the multinormal (g1 = φ1), Ek(φ1) = k(k + 2)/a2k and

Dk(φ1) = k/ak, so that κk(φ1) = 0.

The pseudo-Gaussian test φ(n)N∗ rejects H0 (at asymptotic level α) when-

ever

Q(n)N∗ :=

1

n

∑

1≤i<i′≤m

(ni + ni′)Q(n)N∗;i,i′ > χ2

(m−1)(k0+1);1−α,(5.9)


where, letting S := n−1∑mi=1 niSi,

Q(n)N∗;i,i′ :=

nini′

ni + ni′

1

2(1 + κk)

×{tr[(S−1(Si − Si′))

2]− κk(k+2)κk +2

tr2[S−1(Si −Si′)]

}.

This test statistic is clearly affine-invariant; the following Theorem (see [19])

summarizes its asymptotic properties, which also are those of φ(n)Schott∗.

Theorem 5.2. Assume that (A) and (B) hold. Then:

(i) Q(n)N∗ is asymptotically chi-square with (m − 1)(k0 + 1) degrees of

freedom under⋃

ϑ∈M(Υ)

⋃g1∈F4

1{P(n)

ϑ;g1}, and [provided that (B) is reinforced

into (B′)] asymptotically noncentral chi-square, still with (m − 1)(k0 + 1)degrees of freedom but with noncentrality parameter

k

(k+ 2)κk(g1) + 2rIIϑ,τ +

1

2(1 + κk(g1))rIIIϑ,τ(5.10)

under P(n)

ϑ+n−1/2ν(n)τ(n);g1, with ϑ ∈M(Υ), ντ := limn→∞ ν(n)τ (n) /∈M(Υ),

g1 ∈ F4a(:= F4

1 ∩Fa), rII

ϑ,τ and rIIIϑ,τ defined in (5.2) and (5.3);

(ii) φ(n)N∗ has asymptotic level α under

⋃ϑ∈M(Υ)

⋃g1∈F4

1{P(n)

ϑ;g1};

(iii) φ(n)N∗ is locally and asymptotically most stringent, still at asymp-

totic level α, for⋃

ϑ∈M(Υ)

⋃g1∈F4

1{P(n)

ϑ;g1} against alternatives of the form

⋃ϑ/∈M(Υ){P

(n)ϑ;φ1

}.

6. Asymptotic relative efficiencies. The asymptotic relative efficiencies

of the rank-based tests φ˜(n)K with respect to φ

(n)N∗ and φ

(n)Schott∗ directly follow

as ratios of noncentrality parameters under local alternatives (see Theo-rems 5.1 and 5.2).

Proposition 6.1. Assume that (A), (B′), (C) and (D) hold, and that

g1 ∈ F4a . Then, the asymptotic relative efficiency of φ

˜(n)K with respect to

φ(n)N∗, when testing P

(n)ϑ;g1

against P(n)

ϑ+n−1/2ν(n)τ(n);g1[ϑ ∈M(Υ) and ντ :=

limn→∞ν(n)τ (n) /∈M(Υ)], is

AREϑ,τ,k,g1(φ˜(n)K /φ

(n)N∗)

= (1− ξ)ARE(scale)k,g1

(φ˜(n)K /φ

(n)N∗) + ξARE

(shape)k,g1

(φ˜(n)K /φ

(n)N∗),


where

ARE(scale)k,g1

(φ˜(n)K /φ

(n)N∗) :=

((k+ 2)κk(g1) + 2)L2k(K,g1)

4kLk(K),(6.1)

ARE(shape)k,g1

(φ˜(n)K /φ

(n)N∗) :=

(1 + κk(g1))J 2k (K,g1)

k(k+2)Jk(K)(6.2)

and ξ := ξϑ,τ,k,g1 ∈ [0,1] is given by

ξϑ,τ,k,g1 := ((k+ 2)κk(g1) + 2)rIIIϑ,τ

× [2k(1 + κk(g1))rII

ϑ,τ + ((k +2)κk(g1) + 2)rIIIϑ,τ ]−1.

The “shape AREs” in (6.2) coincide with those obtained in problemsinvolving shape only—such as testing null hypotheses of the form V =V0

for fixed V0 (see [17]). Proposition 6.1 shows that the AREs, with respectto the pseudo-Gaussian tests of Section 5.2, of the rank tests proposed inSection 5.1 are convex linear combinations of these “shape AREs” and the“scale AREs” in (6.1).

Numerical values of (6.1) and (6.2), for various values of the space dimen-sion k and various radial densities (Student, Gaussian and power-exponential),

are given in Table 1 for the van der Waerden test φ˜(n)vdW, the Wilcoxon test

φ˜(n)K1

, and the Spearman test φ˜(n)K2

(the score functions Ka, a > 0 were de-

fined in Section 2.3). These ARE values are uniformly large (with the ex-ception, possibly, of univariate scale Wilcoxon AREs), particularly so underheavy tails, as often in rank-based inference. Also note that the AREs ofthe proposed van der Waerden tests with respect to the parametric Gaus-sian tests are larger than or equal to one for all distributions considered inTable 1. For pure shape alternatives, [29] has shown that a Chernoff–Savage

property holds, that is, infg1 ARE(shape)k,g1

(φ˜(n)vdW/φ

(n)N∗) = 1. One may wonder

whether this uniform dominance property of van der Waerden tests extendsto the present situation. Although it does for usual distributions, includingall Student and power-exponential ones, the general answer unfortunately isnegative; see Section 4 of [29] for a (pathological) counter example.

Note that Theorem 5.2 clearly shows that φ(n)N∗ and φ

(n)Schott∗ are not efficiency-

robust. Indeed, the noncentrality parameter (5.10) under Student radialdensities with 4 + δ degrees of freedom tends to zero as δ → 0, so thatasymptotic local powers are arbitrarily close to the nominal level α. Theefficiency-robustness of our rank tests, quite on the contrary, is not affected,as the ARE values (6.1) and (6.2), under the same Student densities with4 + δ degrees of freedom, both tend to infinity as δ→ 0.


Table 1AREs, for ξ = 0 (pure scale alternatives) and ξ = 1 (pure shape alternatives), of the vander Waerden (vdW), Wilcoxon (W), and Spearman (SP) rank-based tests with respect tothe pseudo- Gaussian tests, under k-dimensional Student (with 5, 8 and 12 degrees offreedom), Gaussian, and power-exponential densities (with parameter η = 2,3,5), for

k = 2, 3, 4, 6, 10 and k→∞

Underlying density

k ξ t5 t8 t12 N e2 e3 e5

vdW 1 0 2.321 1.230 1.082 1.000 1.151 1.376 1.8221 —— —— —— —— —— —— ——

2 0 2.551 1.280 1.102 1.000 1.115 1.296 1.6691 2.204 1.215 1.078 1.000 1.129 1.308 1.637

3 0 2.732 1.322 1.120 1.000 1.092 1.241 1.5581 2.270 1.233 1.086 1.000 1.108 1.259 1.536

4 0 2.881 1.358 1.136 1.000 1.077 1.202 1.4751 2.326 1.249 1.093 1.000 1.093 1.223 1.462

6 0 3.108 1.416 1.163 1.000 1.057 1.151 1.3611 2.413 1.275 1.106 1.000 1.072 1.174 1.363

10 0 3.403 1.498 1.204 1.000 1.037 1.099 1.2391 2.531 1.312 1.126 1.000 1.050 1.121 1.254

∞ 0 4.586 1.894 1.446 1.000 1.000 1.000 1.0001 3.000 1.500 1.250 1.000 1.000 1.000 1.000

W 1 0 1.993 0.939 0.769 0.608 0.519 0.509 0.5171 —— —— —— —— —— —— ——

2 0 2.604 1.185 0.959 0.750 0.694 0.703 0.7431 2.258 1.174 1.001 0.844 0.789 0.804 0.842

3 0 2.929 1.304 1.045 0.811 0.775 0.795 0.8541 2.386 1.246 1.068 0.913 0.897 0.933 1.001

4 0 3.140 1.377 1.096 0.844 0.820 0.844 0.9111 2.432 1.273 1.094 0.945 0.955 1.006 1.095

6 0 3.407 1.467 1.156 0.879 0.866 0.892 0.9611 2.451 1.283 1.105 0.969 1.008 1.075 1.188

10 0 3.685 1.560 1.216 0.908 0.903 0.925 0.9841 2.426 1.264 1.088 0.970 1.032 1.106 1.233

∞ 0 4.323 1.794 1.374 0.955 0.955 0.955 0.9551 2.250 1.125 0.938 0.750 0.750 0.750 0.750

SP 1 0 2.333 1.126 0.935 0.760 0.705 0.724 0.7741 —— —— —— —— —— —— ——

2 0 2.737 1.289 1.063 0.868 0.868 0.924 1.0381 2.301 1.230 1.067 0.934 0.965 1.042 1.168

3 0 2.913 1.348 1.105 0.904 0.924 0.993 1.1361 2.277 1.225 1.070 0.957 1.033 1.141 1.317

4 0 3.016 1.378 1.125 0.920 0.949 1.020 1.1701 2.225 1.200 1.051 0.956 1.057 1.179 1.383

6 0 3.137 1.410 1.142 0.932 0.966 1.032 1.1761 2.128 1.146 1.007 0.936 1.057 1.189 1.414

10 0 3.255 1.438 1.154 0.937 0.969 1.022 1.1391 2.001 1.068 0.936 0.891 1.017 1.144 1.365

∞ 0 3.507 1.503 1.176 0.895 0.895 0.895 0.8951 1.667 0.833 0.694 0.556 0.556 0.556 0.556


7. Simulations. We conducted two simulations, one for pure scale al-ternatives and another one for pure shape alternatives, both in dimensionk = 2. More precisely, starting from two sets of i.i.d. bivariate random vec-tors ε1j (j = 1, . . . , n1 = 100) and ε2j (j = 1, . . . , n2 = 100) with sphericaldensities (the standard bivariate normal and bivariate t-distributions with0.5, 2 and 5 degrees of freedom) centered at 0, we considered independentsamples obtained from

X1j =A1ε1j + θ1, j = 1, . . . , n1,

and

X2j =A2(ℓ)ε2j + θ2, j = 1, . . . , n2,

where A2(ℓ)A′2(ℓ) = (1 + ℓs2)(A1A

′1 + ℓv) [v a symmetric (k × k) matrix

with tr((A1A′1)

−1v) = 0], ℓ= 0,1,2,3. The values of ℓ produce the null (ℓ=0) and increasingly heterogeneous alternatives (ℓ = 1,2,3); all tests beingaffine-invariant, there is no loss of generality in letting A1 = I2 and θ1 =θ2 = 0.

In the first simulation (pure scale alternatives), we generated N = 2,500independent samples, with v = 0 and s2 = 0.30, 0.44, 0.56 and 1.50 underGaussian, t5, t2 and t0.5 alternatives, respectively; these values of s2 havebeen chosen in order to obtain rejection frequencies of the same order un-der those four densities. In the second simulation (pure shape alternatives),we similarly generated N = 2,500 independent samples, with s2 = 0 and

◦vechv = (0,0.18)′ , (0,0.20)′ , (0,0.21)′ and (0,0.22)′ under Gaussian, t5, t2and t0.5 alternatives, respectively, still with the same objective of obtainingcomparable empirical powers under the various densities considered.

In each of these samples, we performed the following eight tests (all at

asymptotic level α = 5%): (a) the modified likelihood ratio test φ(n)MLRT;

(b) the parametric Gaussian test φ(n)N (equivalently, Schott’s original test

φ(n)Schott); (c) its pseudo-Gaussian version φ

(n)N∗, based on (5.9) (equivalently,

the modified Schott test φ(n)Schott∗); (d) the van der Waerden test φ

˜(n)vdW [based

on (5.6)]; (e)–(g) tν -score tests φ˜(n)f t1,ν

with ν = 5, 2 and 0.5 [based on (5.7)],

as well as (h) the Spearman test φ˜(n)K2

[based on (5.8)]. It can be checked

that the Wilcoxon test φ˜(n)K1

, for k = 2, coincides with φ˜(n)f t1,2.

Rejection frequencies are reported in Table 2 for pure scale alternatives,and in Table 3 for pure shape alternatives [the corresponding individual con-fidence intervals (for N = 2,500 replications), at confidence level 0.95, have


half-widths 0.0044, 0.0080 and 0.0100, for frequencies of the order of 0.05(0.95), 0.20 (0.80) and 0.50, resp.]. These frequencies indicate that the ranktests, when based on their asymptotic chi-square critical values, are conser-vative and significantly biased at moderate sample size (100 observations ineach group). In order to remedy this, we also implemented versions of eachof the rank procedures based on estimations of the (distribution-free) quan-tile of the test statistic under known location and known common null valueof the shape. These estimations, just as the asymptotic chi-square quantile,are consistent approximations of the corresponding exact quantiles underthe null, and were obtained for each of the five rank tests under consider-ation in (d)–(h) above, as the empirical 0.05-upper quantiles q0.95 of eachrank-based test statistic in a collection of 105 simulated multinormal sam-ples, yielding q0.95 = 7.2117, 7.6351, 7.7473, 7.7636 and 7.6773, respectively.These bias-corrected critical values all are smaller than the asymptotic chi-square one χ2

3;0.95 = 7.8147, so that the resulting tests are uniformly lessconservative than the original ones. The resulting rejection frequencies aregiven in parentheses.

Inspection of Tables 2 and 3 confirms the fact that the parametric Gaus-sian tests φN , contrary to the pseudo-Gaussian ones φN∗, are invalid undernon-Gaussian densities (culminating, under t0.5, with a size of 0.9992). How-ever, even the pseudo-Gaussian tests φN∗, though resisting non-Gaussiandensities with finite fourth-order moments, are collapsing under the heavy-tailed t0.5 and t2 distributions (with power less than 10−4 under t0.5). Insharp contrast with this, all rank-based tests appear to satisfy the 5%probability level constraint. They are conservative in their original versions(particularly so for van der Waerden scores), but reasonably unbiased (forn1 = n2 = 100) after bias-correction. Empirical power rankings are essen-tially consistent with ARE values; in order to allow for meaningful com-parisons under infinite fourth-order moments, we also provide AREs withrespect to the van der Waerden rank test.

APPENDIX

A.1. Proofs of Lemma 4.1, Theorem 5.1 and Corollary 5.1.

Proof of Lemma 4.1. (i) Fix r ∈ {1, . . . ,m}. Clearly, under P(n)ϑ;g1

,

∆˜II ,rϑ;K =∆II ,r

ϑ;K;g1+ oL2(1) iff

m∑

i=1

ni∑

j=1

c(n)ij;rK

(Rij

n+1

)=

m∑

i=1

ni∑

j=1

c(n)ij;rK

(G1k

(dijσ

))+ oL2(1),(A.1)

where c(n)ij;r := n

−1/2i δi,r. For ϑ ∈M(Υ), the Rij ’s are the ranks of the dij/σ’s,

which under P(n)ϑ;g1

are i.i.d. with distribution function G1k. The asymptotic

RANK

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Y27

Table 2Rejection frequencies (out of N = 2,500 replications), under the null and various scale alternatives (see Section 7 for details), of the

Gaussian modified LRT (φMLRT), the parametric Gaussian test (φN ), its pseudo-Gaussian version (φN∗), and the signed-rank van derWaerden (φ

˜vdW), tν-score (φ

˜ft1,ν

, ν = 0.5, 2, 5), Wilcoxon-type (φ

˜K1

) and Spearman-type (φ

˜K2

) tests, respectively. Sample sizes are

n1 = n2 = 100. ARE values are provided with respect to the parametric pseudo-Gaussian (AREN∗) and van der Waerden rank tests(AREvdW); “ND” means “not defined” (this occurs as soon as one the tests involved is not valid under the distribution under

consideration)

ℓ

Test 0 1 2 3 AREN∗ AREvdW

φLRT N 0.0512 0.3168 0.7932 0.9776 1.000 1.000φMLRT 0.0500 0.3100 0.7876 0.9772 1.000 1.000φN 0.0464 0.3008 0.7760 0.9756 1.000 1.000φN∗ 0.0472 0.2944 0.7568 0.9736 1.000 1.000φ

˜vdW 0.0348 (0.0472) 0.2388 (0.2932) 0.6912 (0.7316) 0.9520 (0.9676) 1.000 1.000

φ

˜ft1,5

0.0444 (0.0496) 0.2604 (0.2724) 0.7080 (0.7200) 0.9552 (0.9600) 0.918 0.918

φ

˜ft1,2

= φ

˜K1

0.0516 (0.0524) 0.2180 (0.2248) 0.6360 (0.6404) 0.9004 (0.9016) 0.750 0.750

φ

˜ft1,0.5

0.0476 (0.0492) 0.1224 (0.1248) 0.3252 (0.3260) 0.5692 (0.5724) 0.360 0.360

φ

˜K2

0.0432 (0.0480) 0.2448 (0.2572) 0.6956 (0.7060) 0.9480 (0.9508) 0.868 0.868

φLRT t5 0.3288 0.6308 0.9168 0.9872 ND NDφMLRT 0.3244 0.6260 0.9144 0.9868 ND NDφN 0.3160 0.6208 0.9092 0.9856 ND NDφN∗ 0.0300 0.1896 0.5268 0.7892 1.000 0.392φ

˜vdW 0.0320 (0.0444) 0.2500 (0.2956) 0.7068 (0.7468) 0.9396 (0.9560) 2.551 1.000

φ

˜ft1,5

0.0428 (0.0480) 0.3004 (0.3152) 0.7740 (0.7812) 0.9636 (0.9676) 2.778 1.089

φ

˜ft1,2

= φ

˜K1

0.0488 (0.0512) 0.2916 (0.2980) 0.7456 (0.7520) 0.9528 (0.9544) 2.604 1.021

28

M.HALLIN

AND

D.PAIN

DAVEIN

E

Table 2Continued

ℓ


φ

˜ft1,0.5

0.0512 (0.0516) 0.1824 (0.1848) 0.4916 (0.4972) 0.7556 (0.7612) 1.543 0.605

φ

˜K2

0.0448 (0.0484) 0.3068 (0.3184) 0.7720 (0.7828) 0.9644 (0.9656) 2.737 1.073

φLRT t2 0.8728 0.9164 0.9496 0.9712 ND NDφMLRT 0.8696 0.9156 0.9496 0.9700 ND NDφN 0.8648 0.9120 0.9480 0.9684 ND NDφN∗ 0.0120 0.0300 0.0672 0.1276 ND NDφ

˜vdW 0.0428 (0.0568) 0.1880 (0.2264) 0.5368 (0.5816) 0.7988 (0.8324) ND 1.000

φ

˜ft1,5

0.0536 (0.0592) 0.2532 (0.2644) 0.6592 (0.6704) 0.9000 (0.9072) ND 1.250

φ

˜ft1,2

= φ

˜K1

0.0508 (0.0532) 0.2732 (0.2816) 0.6912 (0.6964) 0.9212 (0.9236) ND 1.333

φ

˜ft1,0.5

0.0496 (0.0500) 0.2116 (0.2136) 0.5404 (0.5468) 0.8128 (0.8144) ND 1.000

φ

˜K2

0.0572 (0.0588) 0.2568 (0.2652) 0.6632 (0.6708) 0.9036 (0.9080) ND 1.250

φLRT t0.5 0.9992 0.9996 0.9996 0.9988 ND NDφMLRT 0.9992 0.9996 0.9996 0.9988 ND NDφN 0.9992 0.9996 0.9988 0.9988 ND NDφN∗ 0 0 0 0 ND NDφ

˜vdW 0.0388 (0.0520) 0.1464 (0.1764) 0.3096 (0.3572) 0.4608 (0.5188) ND 1.000

φ

˜ft1,5

0.0496 (0.0524) 0.2328 (0.2452) 0.5000 (0.5132) 0.6920 (0.7044) ND 1.543

φ

˜ft1,2

= φ

˜K1

0.0508 (0.0528) 0.3076 (0.3136) 0.6404 (0.6448) 0.8276 (0.8316) ND 2.083

φ

˜ft1,0.5

0.0604 (0.0616) 0.3928 (0.3972) 0.7572 (0.7600) 0.9208 (0.9212) ND 2.778

φ

˜K2

0.0488 (0.0524) 0.2136 (0.2228) 0.4728 (0.4840) 0.6672 (0.6792) ND 1.435

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HOMOGENEIT

Y29

Table 3Rejection frequencies (out of N = 2,500 replications), under the null and various shape alternatives (see Section 7 for details), of theGaussian modified LRT (φMLRT), the parametric Gaussian test (φN ), its pseudo-Gaussian version (φN∗), and the signed-rank van derWaerden (φ

˜vdW), tν-score (φ

˜ft1,ν

, ν = 0.5, 2, 5), Wilcoxon-type (φ

˜K1

) and Spearman-type (φ

˜K2

) tests, respectively. Sample sizes are

n1 = n2 = 100. ARE values are provided with respect to the parametric pseudo-Gaussian (AREN∗) and van der Waerden rank tests(AREvdW); “ND” means “not defined” (this occurs as soon as one the tests involved is not valid under the distribution under

consideration)

ℓ


φLRT N 0.0512 0.1564 0.6032 0.9668 1.000 1.000φMLRT 0.0500 0.1532 0.5984 0.9656 1.000 1.000φN 0.0464 0.1484 0.5900 0.9640 1.000 1.000φN∗ 0.0472 0.1444 0.5812 0.9648 1.000 1.000φ

˜vdW 0.0348 (0.0472) 0.1212 (0.1464) 0.5248 (0.5828) 0.9488 (0.9632) 1.000 1.000

φ

˜ft1,5

0.0444 (0.0496) 0.1452 (0.1536) 0.5456 (0.5596) 0.9464 (0.9496) 0.945 0.945

φ

˜ft1,2

= φ

˜K1

0.0516 (0.0524) 0.1364 (0.1392) 0.4928 (0.5004) 0.9272 (0.9276) 0.844 0.844

φ

˜ft1,0.5

0.0476 (0.0492) 0.1120 (0.1140) 0.3996 (0.4036) 0.8356 (0.8388) 0.648 0.648

φ

˜K2

0.0432 (0.0480) 0.1440 (0.1508) 0.5420 (0.5512) 0.9460 (0.9488) 0.934 0.934

φLRT t5 0.3288 0.4632 0.7840 0.9808 ND NDφMLRT 0.3244 0.4600 0.7816 0.9800 ND NDφN 0.3160 0.4512 0.7728 0.9796 ND NDφN∗ 0.0300 0.1020 0.4204 0.8552 1.000 0.454φ

˜vdW 0.0320 (0.0444) 0.1268 (0.1592) 0.5320 (0.5816) 0.9576 (0.9692) 2.204 1.000

φ

˜ft1,5

0.0428 (0.0480) 0.1572 (0.1676) 0.5928 (0.6036) 0.9720 (0.9740) 2.333 1.059

φ

˜ft1,2

= φ

˜K1

0.0488 (0.0512) 0.1608 (0.1632) 0.5876 (0.5916) 0.9684 (0.9692) 2.258 1.024

30

M.HALLIN

AND

D.PAIN

DAVEIN

E

Table 3Continued

ℓ


φ

˜ft1,0.5

0.0512 (0.0516) 0.1376 (0.1388) 0.5088 (0.5132) 0.9312 (0.9332) 1.896 0.860

φ

˜K2

0.0448 (0.0484) 0.1612 (0.1704) 0.5860 (0.5976) 0.9700 (0.9716) 2.301 1.044

φLRT t2 0.8728 0.8912 0.9376 0.9768 ND NDφMLRT 0.8696 0.8892 0.9364 0.9768 ND NDφN 0.8648 0.8864 0.9332 0.9764 ND NDφN∗ 0.0120 0.0224 0.0808 0.2380 ND NDφ

˜vdW 0.0428 (0.0568) 0.1180 (0.1488) 0.4596 (0.5120) 0.9216 (0.9416) ND 1.000

φ

˜ft1,5

0.0536 (0.0592) 0.1488 (0.1560) 0.5460 (0.5572) 0.9576 (0.9616) ND 1.147

φ

˜ft1,2

= φ

˜K1

0.0508 (0.0532) 0.1584 (0.1612) 0.5640 (0.5668) 0.9668 (0.9668) ND 1.185

φ

˜ft1,0.5

0.0496 (0.0500) 0.1508 (0.1524) 0.5212 (0.5256) 0.9412 (0.9420) ND 1.089

φ

˜K2

0.0572 (0.0588) 0.1440 (0.1500) 0.5288 (0.5420) 0.9516 (0.9564) ND 1.111

φLRT t0.5 0.9992 0.9988 0.9992 0.9992 ND NDφMLRT 0.9992 0.9988 0.9992 0.9992 ND NDφN 0.9992 0.9988 0.9992 0.9992 ND NDφN∗ 0 0 0.0004 0.0008 ND NDφ

˜vdW 0.0388 (0.0520) 0.0964 (0.1208) 0.3328 (0.3792) 0.7960 (0.8328) ND 1.000

φ

˜ft1,5

0.0496 (0.0524) 0.1280 (0.1356) 0.4288 (0.4408) 0.8928 (0.9004) ND 1.254

φ

˜ft1,2

= φ

˜K1

0.0508 (0.0528) 0.1396 (0.1440) 0.4840 (0.4880) 0.9360 (0.9380) ND 1.418

φ

˜ft1,0.5

0.0604 (0.0616) 0.1644 (0.1648) 0.5356 (0.5388) 0.9560 (0.9568) ND 1.543

φ

˜K2

0.0488 (0.0524) 0.1208 (0.1272) 0.3968 (0.4064) 0.8624 (0.8704) ND 1.138


equivalence (A.1) thus follows from Hajek’s classical projection result for

linear rank statistics (see, e.g., [32], Chapter 2), since (a) the c(n)ij;r’s are not

all equal and (b)

maxi,j(c(n)ij;r − n−1∑

i,j c(n)ij;r)

2

∑i,j(c

(n)ij;r − n−1

∑i,j c

(n)ij;r)

2= n−1max

(1− λ

(n)r

λ(n)r

,λ(n)r

1− λ(n)r

)

= o(1) as n→∞

(the Noether condition) holds; see the comments after Assumption (B).

Similarly, for the shape part, ∆˜

III ,rϑ;K =∆

III ,rϑ;K;g1

+ oL2(1) under P(n)ϑ;g1

iff

n−1/2r Mk(V)(V⊗2)−1/2

nr∑

j=1

[K

(Rrj

n+1

)−K

(G1k

(drjσ

))]J⊥k vec(UrjU

′rj)

= oL2(1)

[where J⊥k := Ik2 − 1

kJk satisfies Mk(V)(V⊗2)−1/2J⊥k = Mk(V)(V⊗2)−1/2

and is such that J⊥k vec(UrjU

′rj) is exactly centered], or equivalently iff

T(n)r;l :=

m∑

i=1

ni∑

j=1

c(n)ij;r

[K

(Rij

n+ 1

)−K

(G1k

(dijσ

))][J⊥

k vec(UijU′ij)]ℓ

(A.2)= oL2(1),

for all ℓ ∈ {1,2, . . . , k2}, still under P(n)ϑ;g1

. Now,

E[(T(n)r;ℓ )

2] =Cℓ,k

m∑

i=1

ni∑

j=1

(c(n)ij;r)

2E

[(K

(Ri

n+ 1

)−K

(G1k

(diσ

)))2]

where, denoting by Uij,s the sth component of Uij , Cℓ,k = Var[U211,1] =

2(k − 1)/(k2(k + 2)) for ℓ ∈ Lk := {mk + m + 1,m = 0,1, . . . , k − 1} andCℓ,k = Var[U11,1U11,2] = 1/k2 for ℓ /∈ Lk. Here, the Hajek projection resultfor linear signed-rank statistics (see, e.g., [32], Chapter 3) yields (A.2), since

maxi,j(c(n)ij;r)

2/∑

i,j(c(n)ij;r)

2 = n−1r = o(1), as n→∞.

As for (ii), the result straightforwardly follows, under P(n)ϑ;g1

with ϑ ∈M(Υ), from the multivariate CLT. The result under local alternatives is

obtained as usual, by establishing the joint normality under P(n)ϑ;g1

of ∆ϑ;K;g1

and Λ(n)

ϑ+n−1/2ν(n)τ/ϑ;g1, then applying Le Cam’s third lemma; the required

joint normality follows from a routine application of the classical Cramer–Wold device. �


Proof of Theorem 5.1. (i) The continuity of the mapping ϑ 7→ (PII

ϑ;K ,

PIII

ϑ;K), Proposition 4.2 (jointly with Assumption (D1) and the fact that

[Im −C(n)](Λ(n))−11m = 0), and Lemma 4.1(i), entail

Q˜(n)K = (∆

˜II

ϑ;K)′PII

ϑ;K∆˜II

ϑ;K + (∆˜

III

ϑ;K)′PIII

ϑ;K∆˜

III

ϑ;K + oP(1)

(A.3)= (∆II

ϑ;K;g1)′PII

ϑ;K∆II

ϑ;K;g1 + (∆III

ϑ;K;g1)′PIII

ϑ;K∆III

ϑ;K;g1 + oP(1)

under P(n)ϑ;g1

, ϑ ∈M(Υ) (and therefore, also under the contiguous sequence

P(n)

ϑ+n−1/2ν(n)τ(n);g1). Now, since (ΓII

ϑ;K)1/2PII

ϑ;K(ΓII

ϑ;K)1/2 is a symmetric idem-

potent matrix with rank m− 1, it follows from Lemma 4.1(ii) that the firstterm in (A.3) is asymptotically chi-square with m− 1 degrees of freedom

under P(n)ϑ;g1

, ϑ ∈M(Υ), and asymptotically noncentral chi-square, still withm− 1 degrees of freedom, but with noncentrality parameter

(Lk(K,g1)

4σ4

)2

limn→∞

{(τ (n)II

)′PII

ϑ;Kτ(n)II

}(A.4)

under P(n)

ϑ+n−1/2ν(n)τ(n);g1. Evaluation of the limit in (A.4) yields the first

term in (5.4).As for the shape part, using again Lemma 4.1(ii) and the fact that

(ΓIII

ϑ;K)1/2 ×PIII

ϑ;K(ΓIII

ϑ;K)1/2 is symmetric and idempotent with rank k0(m−1), we obtain similarly that the second term in (A.3) is asymptotically chi-

square with k0(m − 1) degrees of freedom under P(n)ϑ;g1

, ϑ ∈ M(Υ), and

asymptotically noncentral chi-square, still with k0(m − 1) degrees of free-dom but with noncentrality parameter

(Jk(K,g1))2 limn→∞

{(τ (n)III

)′[Im ⊗Hk(V)]PIII

ϑ;K [Im ⊗Hk(V)]τ(n)III

}(A.5)

under P(n)

ϑ+n−1/2ν(n)τ(n);g1. Evaluation of the limit in (A.5) yields the second

term in (5.4). As the two terms in (A.3) are asymptotically uncorrelated [seeLemma 4.1(ii) again], they can indeed be treated separately.

(ii) The fact that φ˜(n)K has asymptotic level α directly follows from the

asymptotic null distribution in part (i) and the classical Helly–Bray theorem.(iii) Optimality is a consequence of the asymptotic equivalence (A.3),

under g1 = f1(∈ Fa), of Q˜(n)f1

and the locally asymptotically optimal test

statistic QΥ, as described in (4.3). �

Proof of Corollary 5.1. (i) Fix g1 ∈ Fa, with Lk(K,g1) 6= 0 6=Jk(K,g1). Clearly, φ

˜(n)K is consistent under P

(n)

ϑ+n−1/2ν(n)τ(n);g1, ϑ ∈M(Υ)


iff the corresponding noncentrality parameter in (5.4) is nonzero. Assumethe latter is zero. Then, the assumptions on g1 imply that s2i /

√λi = s2i′/

√λi′

and

tr

[(V−1/2

(vi√λi

− vi′√λi′

)V−1/2

)2],(A.6)

for all (i, i′). Now, since tr(A2) = 0 implies that A = 0 for any symmetrick× k matrix A, it follows from (A.6) that vi/

√λi = vi′/

√λi′ for all (i, i′).

This is possible only for ντ ∈M(Υ), which establishes the result.(ii) Going back to the definition of g1 7→ Jk(K,g1), we have

Jk(K,g1) =

∫ ∞

0K(G1k(r))rϕg1(r)g1k(r)dr

=1

µk−1;g1

∫ ∞

0K(G1k(r))(−g1(r))r

k dr.

Integrating by parts, Jk(K,g1) =∫∞0 [kK(G1k(r)) + K ′(G1k(r))rg1k(r)] ×

g1k(r)dr = k2 +∫∞0 K ′(G1k(r))r(g1k(r))

2 dr, so that∫∞0 K ′(G1k(r)) ×

r(g1k(r))2 dr > 0 guarantees that Lk(K,g1) = Jk(K,g1)− k2 > 0. The result

follows from part (i) of the corollary. �

A.2. Proof of Proposition 4.2. Consider an arbitrary value ϑ= (ϑ′I ,ϑ

′II,

ϑ′III )

′ = (θ′1, . . . ,θ

′m, σ21′m,1′m⊗ (

◦vechV)′)′ ∈M(Υ) of the parameter and a

(bounded) sequence of corresponding local perturbations ϑ(n) := ϑ +n−1/2ν(n) × τ (n), where

τ (n) = (τ(n)′I ,τ

(n)′II

,τ(n)′III

)′

= (t(n)′1 , . . . , t(n)′m , s

2(n)1 , . . . , s2(n)m , (

◦vechv

(n)1 )′, . . . , (

◦vechv(n)

m )′)′

is such that ϑ(n) ∈M(Υ) for all n. To prove Proposition 4.2, it is sufficient

to show that, under P(n)ϑ;g1

(where g1 is as in Proposition 4.2),

∆˜

II

ϑ(n);K

−∆˜

II

ϑ;K +Lk(K,g1)

4σ4τ(n)II

and

(A.7)∆˜

III

ϑ(n);K

−∆˜

III

ϑ;K +Jk(K,g1)[Im ⊗Hk(V)]τ(n)III

are oP(1) as n → ∞, since the local asymptotic discreteness of ϑ (see,

e.g., [25], Lemma 4.4) allows to replace the nonrandom quantity ϑ(n) with

the random one ϑ in (A.7). Note that ϑ being constrained allows us to re-

strict to ϑ(n) ∈M(Υ). Looking at block i (i ∈ {1, . . . ,m}), Proposition 4.2thus is a corollary of the following result.


Proposition A.1. Assume that (A), (B) and (C) hold, and that g1 ∈Fa. Fix ϑ ∈ M(Υ) and a sequence ϑ(n) ∈ M(Υ) as above. Then, for alli= 1, . . . ,m,

∆˜II ,i

ϑ(n);K

−∆˜II ,i

ϑ;K

+Lk(K,g1)

4σ4s2(n)i and(A.8)

∆˜

III ,i

ϑ(n);K

−∆˜III ,i

ϑ;K

+Jk(K,g1)Hk(V)(◦

vechv(n)i )

are oP(1) under P(n)ϑ;g1

, as n→∞.

Proof. In this proof, we let θni := θi+n

−1/2i t

(n)i , Vn :=V+n

−1/2i v

(n)i ,

and σ2n := σ2 +n

−1/2i s

2(n)i [since ϑ,ϑ(n) ∈M(Υ), σ2

n and Vn do not depend

on i, which explains the notation]. Accordingly, let Z0ij :=V−1/2(Xij − θi),

d0ij := ‖Z0ij‖, U0

ij := Z0ij/d

0ij , Zn

ij := (Vn)−1/2(Xij − θni ), dnij := ‖Zn

ij‖, andUn

ij := Znij/d

nij . Following an argument that goes back to [24], consider the

following truncation of the score function K: for all ℓ ∈N0, define

K(ℓ)(u) :=K

(2

ℓ

)ℓ

(u− 1

ℓ

)I[1/ℓ<u≤2/ℓ] +K(u)I[2/ℓ<u≤1−2/ℓ]

+K

(1− 2

ℓ

)ℓ

((1− 1

ℓ

)− u

)I[1−2/ℓ<u≤1−1/ℓ],

where IA denotes the indicator of A. Since u 7→K(u) is continuous, the func-tions u 7→K(ℓ)(u) are also continuous on (0,1). It follows that the truncatedscores K(ℓ) are bounded for all ℓ. Clearly, it can safely be assumed that K ismonotone increasing (rather than the difference of two monotone increasingfunctions), so that there exists some L such that |K(ℓ)(u)| ≤ |K(u)| for allu ∈ (0,1) and all ℓ≥ L.

We start with the proof that the scale part of (A.8) is oP(1) under P(n)ϑ;g1

.For the shape part, the result is an easy extension of the correspondingresult in [14]; details are left to the reader. Lemma 4.1(i) shows that ∆

˜II ,iϑ;K −

∆II ,iϑ;K;g1

is oP(1), under P(n)ϑ;g1

. Similarly, ∆˜II ,i

ϑ(n);K

−∆II ,i

ϑ(n);K;g1

is oP(1) under

P(n)

ϑ(n);g1

—hence, from contiguity, also under P(n)ϑ;g1

. Consequently, (A.8) is

asymptotically equivalent, under P(n)ϑ;g1

, to

∆II ,i

ϑ(n);K;g1

−∆II ,iϑ;K;g1

+Lk(K,g1)

4σ4s2(n)i .(A.9)

Now, 12n

−1/2i

∑nij=1(K(G1k(d

nij/σn))−k), under P

(n)

ϑ(n);g1

, is asymptotically

normal as n→ ∞, with mean zero and variance 14Lk(K), so that 1

2 (1σ2n−


1σ2 )n

−1/2i

∑nij=1(K(G1k(d

nij/σn))− k) is oP(1), as n→∞, under P

(n)

ϑ(n);g1

, as

well as under P(n)ϑ;g1

(from contiguity). Consequently, (A.9) is asymptotically

equivalent, under P(n)ϑ;g1

, to

C(n)i :=

1

2σ2n−1/2i

ni∑

j=1

(K(G1k(dnij/σn))− k)

− 1

2σ2n−1/2i

ni∑

j=1

(K(G1k(d0ij/σ))− k) +

Lk(K,g1)

4σ4s2(n)i .

Decompose C(n)i into C

(n)i =D

(n;ℓ)i1 +D

(n;ℓ)i2 −R

(n;ℓ)i1 +R

(n;ℓ)i2 +R

(n;ℓ)i3 where,

denoting by E0 expectation under P(n)ϑ;g1

,

D(n;ℓ)i1 :=

1

2σ2n−1/2i

ni∑

j=1

(K(ℓ)(G1k(dnij/σn))−E[K(ℓ)(U)])

− 1

2σ2n−1/2i

ni∑

j=1

(K(ℓ)(G1k(d0ij/σ))−E[K(ℓ)(U)])

− 1

2σ2n−1/2i E0

[ni∑

j=1


],

D(n;ℓ)i2 :=

1

2σ2n−1/2i E0

[ni∑

j=1


]

+Lk(K

(ℓ), g1)

4σ4s2(n)i ,

R(n;ℓ)i1 :=

1

2σ2n−1/2i

ni∑

j=1

[(K(G1k(d0ij/σ))− k)

− (K(ℓ)(G1k(d0ij/σ))−E[K(ℓ)(U)])],

R(n;ℓ)i2 :=

1

2σ2n−1/2i

ni∑

j=1

[(K(G1k(dnij/σn))− k)

− (K(ℓ)(G1k(dnij/σn))−E[K(ℓ)(U)])],

and

R(n;ℓ)i3 :=

1

4σ4(Lk(K,g1)−Lk(K

(ℓ), g1))s2(n)i .

We prove that C(n)i = oP(1) [thus completing the proof that (A.8) is oP(1)

under P(n)ϑ;g1

] by establishing that D(n;ℓ)i1 and D

(n;ℓ)i2 are oP(1) under P

(n)ϑ;g1

,


as n→ ∞, for fixed ℓ, and that R(n;ℓ)i1 , R

(n;ℓ)i2 and R

(n;ℓ)i3 are oP(1) under

the same sequence of hypotheses, as ℓ →∞, uniformly in n. For the sakeof convenience, these three results are treated separately (Lemmas A.1, A.2and A.3).

Lemma A.1. For any fixed ℓ, E0[|D(n;ℓ)i1 |2] = o(1) as n→∞.

Lemma A.2. For any fixed ℓ, D(n;ℓ)i2 = o(1) as n→∞.

Lemma A.3. As ℓ→∞, uniformly in n, (i) R(n;ℓ)i1 is oP(1) under P

(n)ϑ;g1

,

(ii) R(n;ℓ)i2 is oP(1) under P

(n)ϑ;g1

for n sufficiently large and (iii) R(n;ℓ)i3 is

o(1).

Proof of Lemma A.1. First note that D(n;ℓ)i1 = 1

2σ2n−1/2i

∑nij=1[T

(n;ℓ)ij −

E0[T(n;ℓ)ij ]], whereT

(n;ℓ)ij :=K(ℓ)(G1k(d

nij/σn))−K(ℓ)(G1k(d

0ij/σ)), j = 1, . . . , ni

are i.i.d. Writing Var0 for variances under P(n)ϑ;g1

,

E0[|D(n;ℓ)i1 |2] = Var0[D

(n;ℓ)i1 ] =

1

4σ4Var0[T

(n;ℓ)i1 ]≤ 1

4σ4E0[|T(n;ℓ)

i1 |2],

and it only remains to show that, as n→∞,

E0[|T(n;ℓ)i1 |2] = E0[(K

(ℓ)(G1k(dni1/σn))−K(ℓ)(G1k(d

0i1/σ)))

2](A.10)

= o(1)

Now, |dni1/σn − d0i1/σ| ≤ |dni1 − d0i1|/σn + |σ−1n − σ−1|d0i1 is oP(1) under P

(n)ϑ;g1

since |dni1−d0i1| is; see Lemma A.1 in [14]. This and the continuity ofK(ℓ)◦G1k

imply that K(ℓ)(G1k(dni1/σn)) − K(ℓ)(G1k(d

0i1/σ)) = oP(1) under P

(n)ϑ;g1

, as

n→∞. Since K(ℓ) is bounded, this convergence also holds in quadraticmean, which entails (A.10). �

Proof of Lemma A.2. Letting

B(n;ℓ)i1 :=

1

2σ2n−1/2i

ni∑

j=1

(K(ℓ)(G1k(d0ij/σ))−E[K(ℓ)(U)])

one can show that, under P(n)ϑ;g1

, as n→∞,

B(n;ℓ)i1

L−→N(0,

1

4σ4Var[K(ℓ)(U)]

).(A.11)


Under the sequence of local alternatives P(n)

ϑ(n);g1

, as n→∞,

B(n;ℓ)i1 − Lk(K

(ℓ), g1)

4σ4s2(n)i

L−→N(0,

1

4σ4Lk(K

(ℓ))

).

DefiningB(n;ℓ)i2 := 1

2σ2n−1/2i

∑nij=1(K

(ℓ)(G1k(dnij/σn))−E[K(ℓ)(U)]), it follows

from ULAN that, under P(n)ϑ;g1

, as n→∞,

B(n;ℓ)i2 +

Lk(K(ℓ), g1)

4σ4s2(n)i

L−→N(0,

1

4σ4Lk(K

(ℓ))

).(A.12)

Now, from (A.11) and the fact that, under P(n)ϑ;g1

, D(n;ℓ)i1 =B

(n;ℓ)i2 −B

(n;ℓ)i1 −

E0[B(n;ℓ)i2 ] = oP(1) (Lemma A.1), we obtain that, under P

(n)ϑ;g1

, as n→∞,

B(n;ℓ)i2 −E0[B

(n;ℓ)i2 ]

L−→N(0,

1

4σ4Lk(K

(ℓ))

).(A.13)

The result then follows from comparing (A.12) and (A.13). �

We now complete the proof that (A.8) is oP(1) under P(n)ϑ;g1

by provingLemma A.3.

Proof of Lemma A.3. (i) In view of the independence under P(n)ϑ;g1

of

the d0ij ’s,

E0[|R(n;ℓ)i1 |2] = n−1

i

4σ4

ni∑

j=1

E0[[(K(G1k(d0ij/σ))− k)

− (K(ℓ)(G1k(d0ij/σ))−E[K(ℓ)(U)])]2]

=1

4σ4Var[K(U)−K(ℓ)(U)](A.14)

≤ 1

4σ4E[(K(U)−K(ℓ)(U))2]

=1

4σ4

∫ 1

0[K(u)−K(ℓ)(u)]2 du

for all n. Clearly, K(ℓ)(u) converges to K(u), for all u ∈ (0,1). Also, since|K(ℓ)(u)| is bounded by |K(u)|, for all ℓ ≥ L, the integrand in (A.14) isbounded (uniformly in ℓ) by 4|K(u)|2, which is integrable on (0,1). The

Lebesgue dominated convergence theorem thus implies that E0[|R(n;ℓ)i1 |2] =

o(1), as ℓ→∞, uniformly in n.


(ii) Claim (ii) is the same as (i), with dnij/σn replacing d0ij/σ. Accord-

ingly, (ii) holds under P(n)

ϑ(n);g1

. That it also holds under P(n)ϑ;g1

follows from

Lemma 3.5 in [24].(iii) Note that |Lk(K,g1) − Lk(K

(ℓ), g1)|2 = |Cov[K(U) − K(ℓ)(U),Kg1(U)]|2 ≤Lk(g1)×Var[K(U)−K(ℓ)(U)], which is o(1) as ℓ→∞ [see (i)

above]. The result then follows from the boundedness of (s2(n)i ). �

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E.C.A.R.E.S., Institut de Recherche en StatistiqueandDepartement de MathematiqueUniversite Libre de BruxellesCampus de la Plaine CP 210

B-1050 BruxellesBelgiumE-mail: [email protected]

[email protected]: http://homepages.ulb.ac.be/˜dpaindav





mailto:[email protected]

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arxiv.orgarXiv:0806.2963v1 [math.ST] 18 Jun 2008 The Annals of Statistics 2008, Vol. 36, No. 3, 1261–1298 DOI: 10.1214/07-AOS508 c Institute of Mathematical Statistics, 2008 OPTIMAL